{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T20:04:21Z","timestamp":1774296261881,"version":"3.50.1"},"reference-count":39,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,11,25]],"date-time":"2024-11-25T00:00:00Z","timestamp":1732492800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,11,25]],"date-time":"2024-11-25T00:00:00Z","timestamp":1732492800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2025,2]]},"DOI":"10.1007\/s40314-024-02926-w","type":"journal-article","created":{"date-parts":[[2024,11,25]],"date-time":"2024-11-25T20:29:51Z","timestamp":1732566591000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["An efficient modified conjugate gradient parameter for solving the system of symmetric nonlinear equations with application in motion control of coplanar robot"],"prefix":"10.1007","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7262-490X","authenticated-orcid":false,"given":"Jamilu","family":"Sabi\u2019u","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3087-4310","authenticated-orcid":false,"given":"Rana Z.","family":"Al-Kawaz","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,11,25]]},"reference":[{"key":"2926_CR1","doi-asserted-by":"publisher","DOI":"10.1002\/cmm4.1065","volume":"1","author":"AB Abubakar","year":"2019","unstructured":"Abubakar AB, Kumam P, Awwal AM (2019) An inexact conjugate gradient method for symmetric nonlinear equations. Comput Math Methods 1:e1065","journal-title":"Comput Math Methods"},{"key":"2926_CR2","doi-asserted-by":"publisher","unstructured":"Abubakar AB, Mohammad H, Kumam P, Rano SA, Ibrahim AH, Kiri AI (2022) On the new spectral conjugate gradient-type method for monotone nonlinear equations and signal recovery. Math Methods Appl Sci. https:\/\/doi.org\/10.1002\/mma.8479","DOI":"10.1002\/mma.8479"},{"key":"2926_CR3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1155\/2019\/6378368","volume":"2019","author":"HI Ahmed","year":"2019","unstructured":"Ahmed HI, Al-Kawaz RZ, Al-Bayati AY (2019) Spectral three-term constrained conjugate gradient algorithm for function minimizations. J Appl Math 2019:1\u20136","journal-title":"J Appl Math"},{"key":"2926_CR4","doi-asserted-by":"crossref","unstructured":"Al-Kawaz RZ, Al-Bayati AY (2021a) Show off the efficiency of the Dai-Liao method in merging technology for monotonous non-linear problems. Indones J Electr Eng Comput Sci 21(1):505\u2013515","DOI":"10.11591\/ijeecs.v21.i1.pp505-515"},{"key":"2926_CR5","doi-asserted-by":"crossref","unstructured":"Al-Kawaz RZ, Al-Bayati AY (2021b) An effective new iterative CG-method to solve unconstrained non-linear optimization issues. TELKOMN (Telecomm Comput Electr Control) 19(6):1847\u20131856","DOI":"10.12928\/telkomnika.v19i6.19791"},{"key":"2926_CR6","doi-asserted-by":"crossref","unstructured":"Babaie-Kafaki S, Ghanbari R (2014) A descent extension of the Polak\u2013Ribi\u00e9re\u2013Polyak conjugate gradient method. Comput Math Appl 68:2005\u20132011","DOI":"10.1016\/j.camwa.2014.09.019"},{"key":"2926_CR7","doi-asserted-by":"crossref","unstructured":"Babaie-Kafaki S, Ghanbari R (2017) An optimal extension of the Polak\u2013Ribi\u00e9re\u2013Polyak conjugate gradient method. Numer Funct Anal Optim 38:1115\u20131124","DOI":"10.1080\/01630563.2017.1320673"},{"key":"2926_CR8","doi-asserted-by":"publisher","first-page":"1109","DOI":"10.1007\/s11075-017-0299-8","volume":"76","author":"WL Cruz","year":"2017","unstructured":"Cruz WL (2017) A spectral algorithm for large-scale systems of nonlinear monotone equations. Numer Algorithms 76:1109\u20131130","journal-title":"Numer Algorithms"},{"key":"2926_CR9","doi-asserted-by":"publisher","first-page":"1429","DOI":"10.1090\/S0025-5718-06-01840-0","volume":"75","author":"WL Cruz","year":"2006","unstructured":"Cruz WL, Mart\u00ednez J, Raydan M (2006) Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Math Comput 75:1429\u20131448","journal-title":"Math Comput"},{"key":"2926_CR10","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1007\/s002450010019","volume":"43","author":"YH Dai","year":"2001","unstructured":"Dai YH, Liao LZ (2001) New conjugacy conditions and related nonlinear conjugate gradient methods. Appl Math Optim 43:87\u2013101","journal-title":"Appl Math Optim"},{"key":"2926_CR11","doi-asserted-by":"publisher","DOI":"10.1088\/1742-6596\/1366\/1\/012099","volume":"1366","author":"MK Dauda","year":"2019","unstructured":"Dauda MK, Mamat M, Mohamad FS, Magaji AS, Waziri MY (2019) Derivative-free conjugate gradient method via Broyden\u2019s update for solving symmetric systems of nonlinear equations. J Phys Conf Ser 1366:012099","journal-title":"J Phys Conf Ser"},{"key":"2926_CR12","doi-asserted-by":"publisher","first-page":"549","DOI":"10.1090\/S0025-5718-1974-0343581-1","volume":"28","author":"JE Dennis","year":"1974","unstructured":"Dennis JE, Mor\u00e9 JJ (1974) A characterization of superlinear convergence and its application to quasi-Newton methods. Math Comput 28:549\u2013560","journal-title":"Math Comput"},{"issue":"2002","key":"2926_CR13","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1007\/s101070100263","volume":"91","author":"ED Dolan","year":"2002","unstructured":"Dolan ED, Mor\u00e9 JJ (2002) Benchmarking optimization software with performance profiles. Math Program 91(2002):201\u2013213","journal-title":"Math Program"},{"key":"2926_CR14","unstructured":"Fan JY (2003) A modified Levenberg\u2013Marquardt algorithm for singular system of nonlinear equations. J Comput Math 21:625\u2013636"},{"key":"2926_CR15","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1017\/S0334270000005208","volume":"28","author":"A Griewank","year":"1986","unstructured":"Griewank A (1986) The \u2018global\u2019 convergence of Broyden-like methods with a suitable line search. J Aust Math Soc Ser B 28:75\u201392","journal-title":"J Aust Math Soc Ser B"},{"key":"2926_CR16","doi-asserted-by":"publisher","first-page":"1763","DOI":"10.1137\/S0036142901397423","volume":"40","author":"GZ Gu","year":"2002","unstructured":"Gu GZ, Li DH, Qi L, Zhou SZ (2002) Descent directions of quasi-Newton methods for symmetric nonlinear equations. SIAM J Numer Anal 40:1763\u20131774","journal-title":"SIAM J Numer Anal"},{"key":"2926_CR17","first-page":"35","volume":"2","author":"WW Hager","year":"2006","unstructured":"Hager WW, Zhang H (2006) A survey of nonlinear conjugate gradient methods. Pac J Optim 2:35\u201358","journal-title":"Pac J Optim"},{"key":"2926_CR18","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1155\/2020\/3615208","volume":"2020","author":"ET Hamed","year":"2020","unstructured":"Hamed ET, Al-Kawaz RZ, Al-Bayati AY (2020) New investigation for the Liu-story scaled conjugate gradient method for nonlinear optimization. J Math 2020:1\u201312","journal-title":"J Math"},{"key":"2926_CR19","doi-asserted-by":"crossref","unstructured":"Li D, Fukushima M (1999) A globally and superlinearly convergent Gauss\u2013Newton-based BFGS method for symmetric nonlinear equations. SIAM J Numer Anal 37:152\u2013172","DOI":"10.1137\/S0036142998335704"},{"key":"2926_CR20","doi-asserted-by":"publisher","first-page":"71","DOI":"10.3934\/naco.2011.1.71","volume":"1","author":"DH Li","year":"2011","unstructured":"Li DH, Wang XL (2011) A modified Fletcher\u2013Reeves-type derivative-free method for symmetric nonlinear equations. Numer Algebra Control Optim 1:71\u201382","journal-title":"Numer Algebra Control Optim"},{"key":"2926_CR21","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1016\/j.cam.2018.05.006","volume":"344","author":"JK Liu","year":"2018","unstructured":"Liu JK, Feng YM (2018) A norm descent derivative-free algorithm for solving large-scale nonlinear symmetric equations. J Comput Appl Math 344:89\u201399","journal-title":"J Comput Appl Math"},{"key":"2926_CR22","doi-asserted-by":"publisher","first-page":"1073","DOI":"10.1287\/opre.26.6.1073","volume":"26","author":"A Perry","year":"1976","unstructured":"Perry A (1976) A modified conjugate gradient algorithm. Oper Res 26:1073\u20131078","journal-title":"Oper Res"},{"key":"2926_CR23","doi-asserted-by":"crossref","unstructured":"Polak E, Ribi\u00e9re G (1969) Note on the convergence of methods of conjugate directions (in French: Note sur la convergence de m\u00e9thodes de directions conjugu\u00e9es). RIRO 3, pp 35\u201343","DOI":"10.1051\/m2an\/196903R100351"},{"key":"2926_CR24","first-page":"50","volume":"5","author":"J Sabi\u2019u","year":"2016","unstructured":"Sabi\u2019u J (2016) Enhanced derivative-free conjugate gradient method for solving symmetric nonlinear equations. Int J Adv Appl Sci 5:50\u201357","journal-title":"Int J Adv Appl Sci"},{"key":"2926_CR25","first-page":"157","volume":"7","author":"J Sabi\u2019u","year":"2017","unstructured":"Sabi\u2019u J (2017) Effective algorithm for solving symmetric nonlinear equations. J Contemp Appl Math 7:157\u2013164","journal-title":"J Contemp Appl Math"},{"key":"2926_CR26","doi-asserted-by":"publisher","first-page":"234","DOI":"10.3390\/sym13020234","volume":"13","author":"J Sabi\u2019u","year":"2021","unstructured":"Sabi\u2019u J, Muangchoo K, Shah A, Abubakar AB, Jolaoso LO (2021) A modified PRP-CG type derivative-free algorithm with optimal choices for solving large-scale nonlinear symmetric equations. Symmetry 13:234","journal-title":"Symmetry"},{"key":"2926_CR27","doi-asserted-by":"publisher","first-page":"1829","DOI":"10.3390\/sym13101829","volume":"13","author":"J Sabi\u2019u","year":"2021","unstructured":"Sabi\u2019u J, Muangchoo K, Shah A, Abubakar AB, Aremu KO (2021) An inexact optimal hybrid conjugate gradient method for solving symmetric nonlinear equations. Symmetry 13:1829","journal-title":"Symmetry"},{"key":"2926_CR28","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1155\/2015\/961487","volume":"2015","author":"MY Waziri","year":"2015","unstructured":"Waziri MY, Sabi\u2019u J (2015) A derivative-free conjugate gradient method and its global convergence for solving symmetric nonlinear equations. Int J Math Math Sci 2015:1\u20138","journal-title":"Int J Math Math Sci"},{"key":"2926_CR29","first-page":"855","volume":"6","author":"MY Waziri","year":"2016","unstructured":"Waziri MY, Sabi\u2019u J (2016) An alternative conjugate gradient approach for large-scale symmetric nonlinear equations. J Math Comput Sci 6:855\u2013874","journal-title":"J Math Comput Sci"},{"key":"2926_CR30","doi-asserted-by":"crossref","unstructured":"Yu G, Guan L, Li G (2008) Global convergence of modified Polak\u2013Ribi\u00e9re\u2013Polyak conjugate gradient methods with sufficient descent property. J Ind Manag Optim 4:565","DOI":"10.3934\/jimo.2008.4.565"},{"key":"2926_CR31","doi-asserted-by":"publisher","first-page":"116","DOI":"10.1016\/j.camwa.2006.12.081","volume":"55","author":"G Yuan","year":"2008","unstructured":"Yuan G, Lu X (2008) A new backtracking inexact BFGS method for symmetric nonlinear equations. Comput Math Appl 55:116\u2013129","journal-title":"Comput Math Appl"},{"issue":"29","key":"2926_CR32","first-page":"225","volume":"3","author":"GL Yuan","year":"2007","unstructured":"Yuan GL, Lu XW, Wei ZX (2007) A modified trust-region method with global convergence for symmetric nonlinear equations. Math Numer Sin 3(29):225\u2013234","journal-title":"Math Numer Sin"},{"key":"2926_CR33","doi-asserted-by":"publisher","first-page":"44","DOI":"10.1016\/j.cam.2008.10.062","volume":"230","author":"G Yuan","year":"2009","unstructured":"Yuan G, Lu X, Wei Z (2009) BFGS trust-region method for symmetric nonlinear equations. J Comput Appl Math 230:44\u201358","journal-title":"J Comput Appl Math"},{"key":"2926_CR34","doi-asserted-by":"crossref","unstructured":"Zhang L, Zhou W, Li DH (2006) A descent modified Polak\u2013Ribi\u00e9re\u2013Polyak conjugate gradient method and its global convergence. IMA J Numer Anal 26:629\u2013640","DOI":"10.1093\/imanum\/drl016"},{"key":"2926_CR35","doi-asserted-by":"publisher","first-page":"314","DOI":"10.1016\/j.cam.2018.08.017","volume":"347","author":"Y Zhang","year":"2019","unstructured":"Zhang Y, He L, Hu C, Guo J, Li J, Shi Y (2019) General four-step discrete-time zeroing and derivative dynamics applied to time-varying nonlinear optimization. J Comput Appl Math 347:314\u2013329","journal-title":"J Comput Appl Math"},{"key":"2926_CR36","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2019.112454","volume":"367","author":"W Zhou","year":"2020","unstructured":"Zhou W (2020) A modified BFGS type quasi-Newton method with line search for symmetric nonlinear equations problems. J Comput Appl Math 367:112454","journal-title":"J Comput Appl Math"},{"key":"2926_CR37","doi-asserted-by":"publisher","first-page":"370","DOI":"10.1080\/01630563.2013.871290","volume":"35","author":"W Zhou","year":"2014","unstructured":"Zhou W, Shen D (2014) An inexact PRP conjugate gradient method for symmetric nonlinear equations. Numer Funct Anal Optim 35:370\u2013388","journal-title":"Numer Funct Anal Optim"},{"key":"2926_CR38","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1007\/s10957-014-0547-1","volume":"164","author":"W Zhou","year":"2015","unstructured":"Zhou W, Shen D (2015) Convergence properties of an iterative method for solving symmetric non-linear equations. J Optim Theory Appl 164:277\u2013289","journal-title":"J Optim Theory Appl"},{"key":"2926_CR39","first-page":"87","volume":"18","author":"B Zou","year":"2016","unstructured":"Zou B, Zhang L (2016) A nonmonotone inexact MFR method for solving symmetric nonlinear equations. AMO 18:87\u201395","journal-title":"AMO"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-024-02926-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-024-02926-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-024-02926-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,4]],"date-time":"2025-03-04T05:17:03Z","timestamp":1741065423000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-024-02926-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,25]]},"references-count":39,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,2]]}},"alternative-id":["2926"],"URL":"https:\/\/doi.org\/10.1007\/s40314-024-02926-w","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11,25]]},"assertion":[{"value":"13 October 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 April 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 August 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 November 2024","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Informed Consent"}}],"article-number":"49"}}